TY - JOUR
T1 - Perfect subtree property for weakly compact cardinals
AU - Hayut, Yair
AU - Müller, Sandra
N1 - Publisher Copyright:
© 2022, The Hebrew University of Jerusalem.
PY - 2023/3
Y1 - 2023/3
N2 - We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.
AB - We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.
UR - http://www.scopus.com/inward/record.url?scp=85142178745&partnerID=8YFLogxK
U2 - 10.1007/s11856-022-2385-4
DO - 10.1007/s11856-022-2385-4
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AN - SCOPUS:85142178745
SN - 0021-2172
VL - 253
SP - 865
EP - 886
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 2
ER -